§2, exercise 5, p.17:
Let $A_1,\dots,A_n$ be events, and define $S_0,S_1,\dots,S_n$ as follows: $S_0=1$ $$\begin{alignedat}{1}\begin{split}S_{r}=\sum_{J_{r}}P\left(A_{k_{1}}\cap\cdots\cap > A_{k_{r}}\right),\end{split} & 1\leq r\leq n\end{alignedat}$$ where the sum is over the unordered subsets $J_{r}=\left[k_{1},\dots,k_{r}\right]$ of $\left\{ 1,\dots,n\right\}$ .
Let $B_m$ be the event in which each of the events $A_1,\dots,A_n$ occurs exactly $m$ times. Show that $$P\left(B_{m}\right)=\sum_{r=m}^{n}\left(-1\right)^{r-m}C_{r}^{m}S_{r}$$
I don't understand what $B_m$ is!? (my understandings lead to wrong result, tested with particular values of $m$, and $n$)
The original Russian texts:
Пусть $B_{m}$ - событие, состоящее в том, что одновременно произойдет в точности $m$ событий из $A_{1},\dots,A_{n}$.
What is the Russian texts actual meaning? (I want to know if there was problem with translation or it's my fault)
I found an answer from a German forum, which said that:
(google translator): This is indeed a bit incomprehensible, because here m * n events are combined into one
but, I don't know how to interpret the question like that.
P/S: I found an answer, my question could be marked as duplicated: The Probability $P_{[m]}$ that exactly $m$ among the $N$ events $A_1,\dots,A_N$ occur simultaneously
Indeed, I found that the combined result of the online translators (Google, Yandex, Bing, PROMT) is closed to the exercise posted in the link above:
machine translation: Let $B_{m}$ be the event, which consists simultaneous occurrence of exactly $m$ events of $A_{1},\dots,A_{n}$
